Fractals

> Dkschaefer
> Mathematics

Fractal Geometry. Fractal. A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales.

The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension. The prototypical example for a fractal is the length of a coastline measured with different length rulers. The shorter the ruler, the longer the length measured, a paradox known as the coastline paradox.
Fractal. Figure 1a.

The Mandelbrot set illustrates self-similarity. As the image is enlarged, the same pattern re-appears so that it is virtually impossible to determine the scale being examined. Figure 1b.